Integration Worksheet4 solution5

In this page integration worksheet4 solution5 we are going to see solution of some practice question from the worksheet of integration.

Question 17

Integrate the following with respect to x, cos 2 x sin 4 x

Solution:

Now we are going to apply a trigonometric formula for 2 sin A cos B. For that we have to multiply and divide it by 2

        = ∫ (1/2) (2 sin 4x cos 2x) dx

2 sin 4x cos 2x = sin (A + B) + sin (A - B)

       = ∫ (1/2) [sin (4x + 2x) + sin (4x - 2x)] dx

       = ∫ (1/2) [sin 6x + sin 2x] dx

       = (1/2) ∫ [sin 6x] dx  + ∫ [sin 2x] dx

       = (1/2) [ -cos 6x/6 - cos 2x/2] + C

       = (-1/2) [(cos 6x)/6 + (cos 2x)/2] + C


Question 18

Integrate the following with respect to x, sin 10 x sin 2x

Solution:

Now we are going to apply a trigonometric formula for 2 sin A sin B. For that we have to multiply and divide it by 2

        = ∫ (1/2) (2 sin 10x sin 2x) dx

2 sin A sin B = cos (A - B) - cos (A + B)

       = ∫ (1/2) [cos (10x-2x) - cos (10x+2x)] dx

       = ∫ (1/2) [cos 8x - cos 12x] dx

       = (1/2)∫[cos 8x] dx - ∫[cos 12x] dx

       = (1/2)[(sin 8x)/8] - [(sin 12x)/12] + C


Question 19

Integrate the following with respect to x, (1 + cos 2x)/sin² 2 x 

Solution:

Now we are going to apply a trigonometric formula for (1 + cos 2x) and sin2x

(1 + cos 2x) = 2 cos²x

sin 2x = 2 sin x cos x

        = ∫ [(1 + cos 2x)/sin² 2 x] dx

       = ∫ [2 cos²x/(2sin x cos x)²] dx

       = ∫ [2 cos²x/(4 sin² x cos² x)] dx

       = ∫ (1/2sin² x) dx

       = (1/2)∫[1/sin² x] dx

       = (1/2)∫cosec² x] dx

       = (1/2)(-cot x) + C

       = (-1/2)(cot x) + C


Question 20

Integrate the following with respect to x, (e^x - 1)² e^(-4 x)

Solution:

First we are going to expand (e^x - 1)² by using the algebraic formula

(a - b)² = a² - 2 a b + b²

(e^x - 1)² = (e^x)² + 1² - 2 e^x

               = (e^2x) + 1 - 2 e^x

(e^x - 1)² e^(-4 x) = [(e^2x) + 1 - 2 e^x]e^(-4 x)

                            = e^(2x-4x) + e^(-4x) - 2 e^(x - 4x)

                            = e^(-2x) + e^(-4x) - 2 e^(- 3x)

        = ∫ [e^(-2x) + e^(-4x) - 2 e^(- 3x)] dx

       = ∫ [e^(-2x)] dx + ∫ e^(-4x) dx - 2 ∫ e^(-3x) dx

       =  [e^(-2x)/(-2)] + e^(-4x)/(-4) - 2 e^(-3x)/(-3) + C

       =  [- e^(-2x)/2] - e^(-4x)/-4 + (2/3) e^(-3x) + C

integration worksheet4 solution5 integration worksheet4 solution5